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Infinitesimal jet spaces of $\text{Bun}_G$ in positive characteristic

Nikolay Grantcharov

Abstract

Given a semisimple reductive group $G$ and a smooth projective curve $X$ over an algebraically closed field $k$ of arbitrary characteristic, let $\text{Bun}_G$ denote the moduli space of principal $G$-bundles over $X$. For a bundle $P\in\text{Bun}_G$ without infinitesimal symmetries, we provide a description of all divided-power infinitesimal jet spaces, $J_P^{n,PD}(\text{Bun}_G)$, of $\text{Bun}_G$ at $P$. The description is in terms of differential forms on $X^n$ with logarithmic singularities along the diagonals and with coefficients in $(\mathfrak{g}_P^*)^{\boxtimes n}$. Furthermore, we show the pullback of these differential forms to the Fulton-Macpherson compactification of the configuration space, $\hat{X}^n$, is an isomorphism. Thus, we relate the two constructions of Beilinson-Drinfeld and Beilinson-Ginzburg, and as a consequence, give a connection between divided-power infinitesimal jet spaces of $\text{Bun}_G$ and the $\mathcal{L}ie$ operad.

Infinitesimal jet spaces of $\text{Bun}_G$ in positive characteristic

Abstract

Given a semisimple reductive group and a smooth projective curve over an algebraically closed field of arbitrary characteristic, let denote the moduli space of principal -bundles over . For a bundle without infinitesimal symmetries, we provide a description of all divided-power infinitesimal jet spaces, , of at . The description is in terms of differential forms on with logarithmic singularities along the diagonals and with coefficients in . Furthermore, we show the pullback of these differential forms to the Fulton-Macpherson compactification of the configuration space, , is an isomorphism. Thus, we relate the two constructions of Beilinson-Drinfeld and Beilinson-Ginzburg, and as a consequence, give a connection between divided-power infinitesimal jet spaces of and the operad.
Paper Structure (19 sections, 18 theorems, 133 equations)

This paper contains 19 sections, 18 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Residues on $X^n$
  4. Local Residue
  5. Global Residue
  6. Jets on $\text{Bun}_G$
  7. Divided power algebras
  8. Divided power jet spaces
  9. Pairing fibers of jets with the coinvariants
  10. The "universal sheaf" of log-differential forms
  11. The universal space controlling jets
  12. Changing the point $x\in X$
  13. Pullback to the Fulton-Macpherson Compactification
  14. Resolution of the diagonal
  15. Construction of the [BG] Sheaf
  16. ...and 4 more sections

Key Result

Theorem 2.1

BZF(Strong Residue Theorem) Let $\mathcal{E}$ be a locally free sheaf over $X$. Then a section $s\in\mathcal{E}(D_x^\times)$ extends to $\tilde{s}\in\mathcal{E}(X\setminus x)$ if and only if As a corollary, there is a perfect pairing, given by the reside at $x$, between

Theorems & Definitions (37)

  • Theorem 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Theorem 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • Corollary 3.4
  • ...and 27 more